Properties of local orthonormal systems Part I: Unconditionality in Lp$L^p$, 1<p<∞$1&lt;p&lt;\infty$

Abstract

Assume that we are given a filtration on a probability space of the form that each is generated by the partition of one atom of into two atoms of having positive measure. Additionally, assume that we are given a finite‐dimensional linear space S of ‐measurable, bounded functions on Ω so that on each atom A of any σ‐algebra , all ‐norms of functions in S are comparable independently of n or A. Denote by the space of functions that are given locally, on atoms of , by functions in S and by the orthoprojector (with respect to the inner product in ) onto . Since satisfies the above assumption and is then the conditional expectation with respect to , for such filtrations, martingales are special cases of our setting. We show in this article that certain convergence results that are known for martingales (or rather martingale differences) are also true in the general framework described above. More precisely, we show that the differences form an unconditionally convergent series and are democratic in for . This implies that those differences form a greedy basis in ‐spaces for .